Differential equations Questions and Answers

A differential equation a point and a slope field are given A slope field or direction field consists of line segments with slopes given by the differential equation These line segments give a visual perspective of the slopes of the solutions of the differential equation dy dx 1 Show Transcribed Text

To answer questions 28 32 use this information to find the unknown parts of the triangle a 32 B 32 B 32 C 71 28 A a 103 b 77 C 13 d 193

Centerville is the headquarters of Greedy Cablevision Inc The cable company is about to expand service to two nearby towns Springfield and Shelbyville There needs to be cable connecting Centerville to both towns The idea is to save on the cost of cable by arranging the cable in a Y shaped configuation Centerville is located at 10 0 in the xy plane Springfield is at 0 3 and Shelbyville is at 0 3 The cable runs from Centerville to some point x 0 on the x axis where it splits into two branches going to Springfield and Shelbyville Find the location x 0 that will minimize the amount of cable between the 3 towns and compute the amount of cable needed Justify your answer To solve this problem we need to minimize the following function of x f x We find that f x has a critical number at x To verify that f x has a minimum at this critical number we compute the second derivative f x and find that its value at the critical number is a positive number Thus the minimum length of cable needed is

Graph the function y 7 3x 14 cos x 0 x 2x by identifying the domain and any symmetries finding the derivatives y and y finding the critical points finding where the curve is increasing and where it is decreasing finding the points of inflection and concavity identifying any asymptotes and plotting any key points such as intercepts critical points and inflection points Then find coordinates of absolute extreme points if any

5 Points Problem 4 Suppose that w x y z g u x y v y z g X Z Show that tiable function of u x y z and v x y z Y Y 0 xwx ywy zw is a differe

2 10 3 Consider the differential equation below and the general solution Verify the solution satisfies the given differential equation y 6y 27y y sin 3x d C

1 12 Solve the differential equation dy 3x y dx 1 dy 3 x y dx 5 xyy x 1 7 e 1 y 2 cosx dp 9 11 13 dt t p p t 1 15 de dt dy dx t sec 0 Oe dA 2 xe y 0 0 8 4 xy y 3 6 y xe 0 dy dx 10 12 dy dx 13 20 Find the solution of the differential equation that satisfies the given initial condition 14 Ab cos br A 0 b dr 16 x y k sec y y 1 6 dz dt dH dR x y4 dP dt 2x y 1 et 0 RH 1 R In H Pt P 1 2 17 du dt 2t sec t 2u 18 x 3y x 1 20 19 x In x y 1 dy dx X sin X y u 0 5 dy dx 0 y y y 0 1 y 0 1 y 1 1 21 Find an equation of the curve that passes th 0 2 and whose slope at x y is x y 22 Find the function f such that f x x x f 0 2 23 Solve the differential equation y x y b change of variable u x y 24 Solve the differential equation xy y x change of variable v y x 25 a Solve the differential equation y 2x b Solve the initial value problem y 2x y 0 0 and graph the solution c Does the initial value problem y P y 0 2 have a solution Explain 2x

9 Find the linearization of f x 25 x at a 3

4 Given the function a Find the points of inflection b Find the intervals of concavity t 12x 24 x 0 f x 12x x 2x 1 0 2 2x 1 x 2 EZ 240 2 1 1 X 1 X 1

At least one of the answers above is NOT correct Identify the marked points as being an absolute maximum or minimum a A N F i

f x 6 cos x on 27 2 Its derivative is f x help formulas The graph that correctly displays f x in black and f x in orange is OA OB OC OD 6 28319 8 28319 WAN AA AM D 6 28 19 28513 28314 8 28319 128311 Using the graphs of fand f indicate where f is increasing and decreasing Enter DNE if su f is increasing on help intervals f is decreasing on

17 Which of the following functions are solutions of the dif ferential equation y y sinx a y sin x b y cos x c y x sin x d y x cos x

Solve the differential equation by variation of parameters 2y 4y 4y e sec x y x C e cos x C e sin x ecos x ln cos x e sin x 2

Find the general solution of the given higher order differential equation d u dt u t d u dt 2u 0

The area of a wound decreases exponentially with time The area A of a wound over time t can be modeled by the exponential equation A A e 05r Where is the initial wound area If the wound area is 6cm how large will the wound area be 14 days O 3 cm 4 cm 2 6 cm 1 5 cm

1 point A company operates two plants which manufacture the same item and whose total cost functions are C 9 0 02g and C 44 0 04 where q and are the quantities produced by each plant The total quantity demanded qq 92 is related to the price p by P 70 0 044 How much should each plant produce in order to maximize the company s profit 91 92 Adapted from M Rosser Basic Mathematics for Economists p 318 New York Routledge 1993

b Solve Ju t Ju 3x x 1 U x u x 0 f x t 0

aluate x 30x 30y dA where D is the shaded region enclosed by the lemniscate curve sin 20 in the figure 0 5 r sin 20 X 0 5 se symbolic notation and fractions where needed

12 For the function below find a the critical nur

Part 3 of 3 Find the open intervals where the function f x 4x 24x 170x 1 is c Select the correct choice below and if necessary fill in the answer box to co A The function has a point of inflection at 2 403 Type an ordered pair using integers or fractions Simplify your answe

21 What is the period of the graph of y sin2x A 4 B C 7 FIN

4 9 Wikipedia A sample survey of 852 adult Internet users found that 53 look for information on the online collaborative en cyclopedia Wikipedia a Give the standard error SE of p the proportion of all adult Internet users who look for information on Wikipedia b Give a 95 confidence interval for the proportion of all adult Internet users who look for information on Wikipedia W wil genius Does cheating enhance creativity Researchers recruited 178 bjects and asked participants to guess whether the outcome of a virtual coin ss would be heads or tails After indicating their prediction participants had to press a button toss the coin virtually They were asked to press the button only once but they were given the portunity to test the button several times before beginning the experiment Thus participants uld cheat by pressing the button before making their prediction and appear to have pressed the itton one time for each prediction Participants then reported whether they had guessed correctly id received a 1 bonus if they had The program recorded the outcomes of the initial virtual in tosses so that researchers could tell whether participants cheated After the coin toss task all rticipants were given a test that measured creativity Here are the summary statistics for scores

Given P x 1 1 W full equation including P P x x 1 x

A small radio transmitter broadcasts in a 25 mile radius If you drive along a straight line from a city 32 miles north of the transmitter to a second city 30 miles east of the transmitter during how much of the drive will you pick up a signal from the transmitter Give your answer rounded to the nearest tenth of a mile

6 12 Determine whether the given function is a solution of the differential equation 6 y sin x cos x y y 2 sin x 7 y e e 2 y 2y 2e

Compute L3 and R3 for Graphs A and B The value of L3 for Graph A is Simplify your answer 16 14 12 10 00 Ay 8 79 6 4 2 0 0 16 14 Ay 12 10 8 6 st 4 NC 2 2 DEPEN y f x 4 6 6 Graph A y g x 8 X 10 X

Which slope field is associated with the given differential equation dy x 1 dx Choose the correct slope field O A 6 4 2 2 46 OB 3 st Ci O C 13 24 16 71 11 6 4 2 246

Give the order of the differential equation below wh xy 6y y 8 Choose the correct answer below OA The given equation is a second order differ higher derivative

sin COS tan 1 0 2 3 ans 1 x t From the information given find the quadrant in which the terminal point determined by t lies Input I II III or IV a sin t 0 and cos t 0 quadrant b sin t 0 and cos t 0 quadrant c sin t 0 and cos t 0 quadrant d sin t 0 and cos t 0 quadrant

Find the general or particular solu 0 2x7 c

Please show all your work NO WORK NO CREDIT 55 44 Find a formula for the exponential function passing through the points 3 and 2 20

ssignmen Previous Problem P point At noon ship A

Find any vertical asymptotes as well as the location of any holes in the graph of the given rational functio x 3x 28 x 7x 12 f x

Question 1 Eliminate the parameter and write the set of parametric equations in rectangular form x 3t 2 y 9t 4 Oy 9x 4 Oy x 10 3 Oy 3x 2 Oy 3x 10 2

21 28 Solve the initial value problem dy dt 21 t 1 t 0 y 6 when t 0

3 2 2 Let g x x 9x 7x and let c be the number that satisfies the Mean Value Theorem for g on the interval 3 x 6 What is c

Use the Chain Rule to find aw dx Typo 11 dw dt 1 t where w cos 12x sin 4y x and y 1 expression using y andy as the variables

Find the integral 7x6dx 3 x7 3 Of 1 4 3 x7 4 7x6 3 x7 C C C 2 3 x O 3 C

48 Solve the given equation 2 sin 0 5 sin 0 12 0 13 7 5 10 6pts Solve the given equation cos 20 cos 0 1 2 SCH 14 7 5 45 6pts Use a double or half angle formula to solve the equati in the interval 0 27 cos 20 cos 0 2

A tank contains 5000 L of pure water Brine that contains 30 g of salt per liter of water is pumped into the tank at a rate of 25 L min C t 30t 200 t A After how many minutes will the salt concentration be 20 grams per liter B State the equation of the horizontal asymptote and explain what it is telling you in this context

Graph the following function an 1 f x 2 3 x 2

3 14 Values of Trigonometric Functions Use an Addition or Subtraction Formula to find the exact value of the expression as demonstrated in Example 1 3 sin 75 5 cos 105 7 tan 15 9 sin 11 tan 7 19T 12 13 cos 5 a 12 11T 12 4 sin 15 6 cos 195 8 tan 165 10 cos 17T 12 12 sin 14 tan 5TT 12 7TT 12

Construct the particular solution to the homogenoeus linear second order ODE d 2x dt2 10 dx dt 50x 0 subject to initial conditions x 0 13 dx dt 0 0 Find the value of x at t 3 5

3 Find the exact length of the following curve a y ln 1 x 0 x 1 1 b y 1 e 2 0 x 1

Solve the initial value problems Answer y e y 2y 15y 0 y 0 1 y 0 2 e 3x e

Solve the given boundary value problem y x y 2y 2y 2x 2 y 0 0 y

Solve the given y x boundary value problem y y x 1 y 0 4 y 1 0

Solve the given initial value problem y x 4y 4y 3y 0 y 0 1 y 0 9

Solve the given initial value problem d y d8 y 8 y 0 y 3 0 y 3 8

Solve the given boundary value problem If an answer does not exist enter DNE y 8y 16y 0 y 0 1 y 1 0 y x